Among classical theorems of model theory, preservation theorems are results that relate the syntactic form of formulas to semantic closure properties of the structures they define. The status of preservation theorems in the finite has been an active area of research in finite model theory. More recent work in the area has shifted the focus from the class of all finite structures to classes of structures satisfying natural structural restrictions. In this talk I will examine various recent results relating to preservation under homomorphisms and extensions, both on the class of all finite structures and on more restricted classes.
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